Single Slit - srjcstaff.santarosa.edusrjcstaff.santarosa.edu/~lwillia2/private41/41ch38.pdf ·...

The diffraction pattern from a hexagonally-shaped hole. Note the six-fold symmetry of the pattern. Observation of such complex patterns can reveal the underlying symmetry structure of the object that diffracts the light. This is the basis for many physical techniques that study patterns in the microscopic world.

Single Slit Interference Is calledDiffraction

Single Slit

Single Slit Interference

Single Slit Diffraction

rΔ

When the path length differs by half a wavelength then the rays will interfere destructively. For Rays 1 & 3:

sin2 2ar λθΔ = =

Single Slit Diffraction

a

Dark fringe: sin 0,1,2,3,...m maλθ = =

Dark fringe: sin 0,1,2,3,...m maλθ = =

Single Slit Problem

A narrow slit is illuminated with sodium yellow light of wavelength 589 nm. If the central maximum extends to ±30°, how wide is the slit?

a. 0.50 mmb. 2.2 × 10–6 mc. 3.3 × 10–5 md. 1.18 µme. 5.89 µm

Dark fringe: sin maλθ =

Dark fringe: sin 0,1,2,3,...m maλθ = =

Width of the Central Maximum

2w y=

Dark fringe: sin 0,1,2,3,...m maλθ = =

If the incident light has λ=690nm (red) and a = 4 x 10-6m, find the width of the central bright fringe.

Single Slit Diffraction

a

Dark fringe: sin 0,1,2,3,...m maλθ = =

Circular Aperature

Dark fringe: sin 0,1, 2,3,...m mDλθ = =

Diffraction of a Penny

Fig 38-3, p.1207

Central Bright Spot: Poisson Spot

Resolution

• The ability of optical systems to distinguish between closely spaced objects is limited because of the wave nature of light

• If two sources are far enough apart to keep their central maxima from overlapping, their images can be distinguished– The images are said to be

resolved• If the two sources are close together,

the two central maxima overlap and the images are not resolved

Fig 38-13, p.1215

Diffraction Resolution

Resolving Power

min

1.22

Rayleigh Criterion

Dλθ =

The diffraction limit: two images are just resolvable when the center of the diffraction pattern of one is directly over the first minimum of

the diffraction pattern of the other.

D: Aperture Diameter

Standing back from a Georges Seurat painting, you can cannot resolve the dots but a camera, at the same distance can. Assume that light enters your eyes through pupils that have diameters of 2.5 mm and

enters the camera through an aperture with diameter of 25 mm.Assume the dots in the painting are separated by 1.5 mm and that the

wavelength of the light is 550 nm in vacuum. Find the distance at which the dots can just be resolved by a) the camera b) the eye.

Georges Seurat’s 1884 pointillist masterpiece

A Sunday on La Grand Jatte

Space Shuttle ResolutionWhat, approximately, are the dimensions of the smallest object on Earth that the astronauts can resolve by eye at 200 km height from the space shuttle? Assume l = 500 nm light and a pupil diameter D = 0.50 cm. Assume eye fluid has an average n = 1.33.

a. 150 mb. 100 mc. 250 md. 25 me. 18 m

Light from a small source passes by the edge of an opaque objectand continues on to a screen. A diffraction pattern consisting of bright and dark fringes appears on the screen in the region above the edge of the object.

Light Intensity, Double Slit

• The interference pattern consists of equally spaced fringes of equal intensity

• This result is valid only if L >> d and for small values of θ

2 2max max

sin cos cosI I Iπd θ πd yλ λL

⎛ ⎞ ⎛ ⎞= ≈⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Single Slit Intensity• The intensity can be expressed as

• Imax is the intensity at θ = 0– This is the central maximum

• Minima occur at

( ) 2

maxsin sin

sin I I

πa θ λπa θ λ

⎡ ⎤= ⎢ ⎥

⎣ ⎦

darkdark

sin or sin πa θ λmπ θ mλ a

= =

Combined EffectsInterference maximum coincides with the first diffraction minimum.

Two-Slit Diffraction Patterns, Maxima and Minima

• To find which interference maximum coincides with the first diffraction minimum

– The conditions for the first interference maximum

• d sin θ = mλ– The conditions for the first

diffraction minimum• a sin θ = λ

sin sin d θ mλ d ma θ λ a

= → =

Intensity of Two-Slit Diffraction

• The broken blue line is the diffraction pattern

• The red-brown curve shows the cos2 term– This term, by itself, would

result in peaks with all the same heights

– The uneven heights result from the diffraction term (square brackets in the equation)

( ) 22

maxsin sin sin cos

sin /

I I/

πa θ λπd θλ πa θ λ

⎡ ⎤⎛ ⎞= ⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦

Interference maximum coincides with the first diffraction minimum.

Diffraction Gratings

Diffraction Grating• The condition for maxima is

– d sin θbright = mλ• m = 0, ±1, ±2, …

• d is the slit spacing• The integer m is the order

number of the diffraction pattern

Dispersion via Diffraction

: sin , 0,1, 2,3constructive d m mθ λ= =

Hydrogen Spectra

Helium Spectra

Mercury Spectra

Neon Spectrum

Oxygen Spectrum

Argon Spectrum

Absorption Spectrum of Hydrogen Gas

•Cosmological Redshift: Expanding Universe•Stellar Motions: Rotations and Radial Motions•Solar Physics: Surface Studies and Rotations•Gravitational Redshift: Black Holes & Lensing•Exosolar Planets via Doppler Wobbler

Spectral lines shift due to the relative motion between the source and the observer

• Red Shift: Moving Away• Blue Shift: Moving Toward

Diffraction Gratings

: sin , 0,1, 2,3constructive d m mθ λ= =

Note: The greater the wavelength, the greater the angle.

A mixture of violet light (410 nm in vacuum) and red light (660 nm in vacuum) fall on a grating that contains 1.0 x104 lines/cm. For each wavelength, find the angle and the distance from the

central maximum to the first order maximum.

Diffraction Grating Problem

White light is spread out into spectral hues by a diffraction grating. If the grating has 1000 lines per cm, at what angle will red light (l = 640 nm) appear in first order?

a. 14.68°b. 7.35°c. 17.73°d. 3.67°e. 1.84°

Resolving Power of a Diffraction Grating

• For two nearly equal wavelengths, λ1 and λ2, between which a diffraction grating can just barely distinguish, the resolving power, R, of the grating is defined as

• Therefore, a grating with a high resolution can distinguish between small differences in wavelength

2 1

λ λRλ λ λ

≡ =− Δ

Resolving Power of a Diffraction Grating

• The resolving power in the mth-order diffraction is R = Nm– N is the number of slits– m is the order number

• Resolving power increases with increasing order number and with increasing number of illuminated slits

Grating Resolution

Determine the number of grating lines necessary to resolve the 589.59 nm and 589.00 nm sodium lines in second order.

a.999b.680c. 500d.340e. 380

Limits of Vision

112.4 10e x mλ −=

ElectronWaves

Electron Diffraction

Electron Microscope

Electron microscope picture of a fly. The resolving power of an optical lens depends on the wavelength of

the light used. An electron-microscope exploits the wave-like properties of particles to reveal details that would be impossible to see

with visible light.

Electron Microscope

Stem Cells

The fossilized shell of a microscopic ocean animal is magnified 392 times its actual size.

Salmonella Bacteria

CrystallographyThe International Union of Crystallography has redefined the term crystal to mean ``any solid having an essentially discrete diffraction pattern.

Diffraction of X-Rays by Crystals

• X-rays are electromagnetic waves of very short wavelength

• Max von Laue suggested that the regular array of atoms in a crystal could act as a three-dimensional diffraction grating for x-rays

• Laue Pattern for Beryl

Diffraction of X-Rays by Crystals, Set-Up

• A collimated beam of monochromatic x-rays is incident on a crystal

• The diffracted beams are very intense in certain directions– This corresponds to

constructive interference from waves reflected from layers of atoms in the crystal

• The diffracted beams form an array of spots known as a Laue pattern

Laue Pattern for Rubisco

X-Ray Diffraction: Bragg’s Law• This is a two-dimensional

description of the reflection of the x-ray beams

• The condition for constructive interference is2d sin θ = mλwhere m = 1, 2, 3

• This condition is known as Bragg’s law

• This can also be used to calculate the spacing between atomic planes

X-Ray Diffraction

In an X-ray diffraction experiment using X-rays of = 0.500 × 10–10 m, a first-order maximum occurred

at 5.00°. Find the crystal plane spacing.a. 2.87 × 10–10 mb. 1.36 × 10–10 mc. 6.24 × 10–9 md. 1.93 × 10–9 me. 5.74 × 10–9 m

Polarization of Light Waves

• The direction of polarization of each individual wave is defined to be the direction in which the electric field is vibrating

• In this example, the direction of polarization is along the y-axis

Unpolarized Light• All directions of vibration

from a wave source are possible

• The resultant em wave is a superposition of waves vibrating in many different directions

• This is an unpolarized wave

• The arrows show a few possible directions of the waves in the beam

Circularly Polarized EM Wave

Elliptically Polarized EM Wave

Polarization of Light

• A wave is said to be linearly polarized if the resultant electric field E vibrates in the same direction at all times at a particular point

• The plane formed by E and the direction of propagation is called the plane of polarization of the wave

Methods of Polarization

• It is possible to obtain a linearly polarized beam from an unpolarized beam by removing all waves from the beam except those whose electric field vectors oscillate in a single plane

• Processes for accomplishing this include– selective absorption– reflection– double refraction– scattering

Polarization of Light

Plane Polarized

Circular Polarized

Polarization upon Reflection

Intensity of Polarized Light, Examples

• On the left, the transmission axes are aligned and maximum intensity occurs

• In the middle, the axes are at 45o to each other and less intensity occurs

• On the right, the transmission axes are perpendicular and the light intensity is a minimum

Polarization by Selective Absorption

• The most common technique for polarizing light• Uses a material that transmits waves whose electric field

vectors lie in the plane parallel to a certain direction and absorbs waves whose electric field vectors are perpendicular to that direction

Malus’ Law2

0 cosI I θ=

PolarizationThe intensity of unpolarized light passing through a polarizer will be reduced my ½, because the average value of the cosine squared term over all directions is ½.

20 cosI I θ=

If the incident beam is linearly polarized along the vertical direction and has an intensity of 50 W/m2 and the angle is 30

degrees, which set up transmits more light? Determine the average intensity of the transmitted beam for both setups.

If the light is unpolarized, which set up transmits the most light?

Polarization Problem

Unpolarized light is passed through three successive Polaroid filters, each with its transmission axis at 45° to the preceding filter. What percentage of light gets through?

a. 0%b. 12.5%c. 25%d. 50%e. 33%

Polarization by Reflection

• When an unpolarized light beam is reflected from a surface, the reflected light may be– Completely polarized– Partially polarized– Unpolarized

• It depends on the angle of incidence– If the angle is 0°, the reflected beam is unpolarized– For other angles, there is some degree of polarization– For one particular angle, the beam is completely polarized

Polarization by Reflection, Partially Polarized Example

• Unpolarized light is incident on a reflecting surface

• The reflected beam is partially polarized

• The refracted beam is partially polarized

Polarization by Reflection, Completely Polarized Example• Unpolarized light is

incident on a reflecting surface

• The reflected beam is completely polarized

• The refracted beam is perpendicular to the reflected beam

• The angle of incidence is Brewster’s angle

Polarization by Reflection• The angle of incidence for which the

reflected beam is completely polarized is called the polarizing angle, θp

• Brewster’s law relates the polarizing angle to the index of refraction for the material

• θp may also be called Brewster’s angle

sin tan

cos p

pp

θn θ

θ= =

Brewster’s Angle

Sunlight reflected from a smooth ice surface is completely polarized. Determine the angle of incidence. (nice = 1.31.)

a.52.6°b.25.6°c. 65.2°d.56.2°e. 49.8°

sin tan

cos p

pp

θn θ

θ= =

Polarization by Double Refraction: Birefringence

• In certain crystalline structures, the speed of light is not the same in all directions

• Such materials are characterized by two indices of refraction

• They are often called double-refracting or birefringent materials

Optical Stress Analysis

• Some materials become birefringent when stressed

• When a material is stressed, a series of light and dark bands is observed– The light bands correspond to

areas of greatest stress• Optical stress analysis uses

plastic models to test for regions of potential weaknesses

Polarization by Scattering

• When light is incident on any material, the electrons in the material can absorb and reradiate part of the light– This process is called scattering

• An example of scattering is the sunlight reaching an observer on the Earth being partially polarized

Polarization by Scattering, cont.

• The horizontal part of the electric field vector in the incident wave causes the charges to vibrate horizontally

• The vertical part of the vector simultaneously causes them to vibrate vertically

• If the observer looks straight up, he sees light that is completely polarized in the horizontal direction

Scattering, cont.

• Short wavelengths (blue) are scattered more efficiently than long wavelengths (red)

• When sunlight is scattered by gas molecules in the air, the blue is scattered more intensely than the red

• When you look up, you see blue• At sunrise or sunset, much of the blue is scattered

away, leaving the light at the red end of the spectrum

Optical Activity

• Certain materials display the property of optical activity– A material is said to be optically active if it

rotates the plane of polarization of any light transmitted through it

– Molecular asymmetry determines whether a material is optically active

LasersLight Amplification by

Simulated Emission

"A splendid light has dawned on me about the absorption and emission of radiation..."

Albert Einstein, 1916

Stimulated Emission

"When you come right down to it, there is really no such thing as truly spontaneous emission; its all stimulated emission. The only distinction to be made is whether the field that does the stimulating is one that you put there or one that God put there..."

David Griffths, Introduction to Quantum Mechanics

Holography

Light is in a Superposition of Polarization States!

This is a Quantum Effect.

Two crossed polarizers cut all light.But add a third polarizer in between andLight shines through! How is this so?

Three Polarizers ParadoxYou have a field with cows. To make sure that now cows get out, you put up two fences. They stay in their field. But you're really paranoid, so you put a third fence in between the two. Now, all of a sudden, one fourth of your cows are wandering in your neighbor's field.

You have a field with cows. To make sure that now cows get out, you put up two fences. They stay in their field. But you're really paranoid, so you put a third fence in between the two. Now, all of a sudden, one fourth of your cows are wandering in your neighbor's field.